As we dive into the digital era, there is growing concern about the amount of personal digital information that is being gathered about us. Websites often track people's browsing behavior, health care insurers gather medical data, and many smartphones and navigation systems store or trans- mit information that makes it possible to track the physical location of their users at any time. Hence, anonymity, and privacy in general, are in- creasingly at stake. Anonymity protocols counter this concern by offering anonymous communication over the Internet. To ensure the correctness of such protocols, which are often extremely complex, a rigorous framework is needed in which anonymity properties can be expressed, analyzed, and ulti- mately verified. Formal methods provide a set of mathematical techniques that allow us to rigorously specify and verify anonymity properties. This thesis addresses the foundational aspects of formal methods for applications in security and in particular in anonymity. More concretely, we develop frameworks for the specification of anonymity properties and propose algorithms for their verification. Since in practice anonymity pro- tocols always leak some information, we focus on quantitative properties which capture the amount of information leaked by a protocol. We start our research on anonymity from its very foundations, namely conditional probabilities - these are the key ingredient of most quantitative anonymity properties. In Chapter 2 we present cpCTL, the first temporal logic making it possible to specify conditional probabilities. In addition, we present an algorithm to verify cpCTL formulas in a model-checking fashion. This logic, together with the model-checker, allows us to specify and verify quantitative anonymity properties over complex systems where probabilistic and nondeterministic behavior may coexist. We then turn our attention to more practical grounds: the constructions of algorithms to compute information leakage. More precisely, in Chapter 3 we present polynomial algorithms to compute the (information-theoretic) leakage of several kinds of fully probabilistic protocols (i.e. protocols with- out nondeterministic behavior). The techniques presented in this chapter are the first ones enabling the computation of (information-theoretic) leak- age in interactive protocols. In Chapter 4 we attack a well known problem in distributed anonymity protocols, namely full-information scheduling. To overcome this problem, we propose an alternative definition of schedulers together with several new definitions of anonymity (varying according to the attacker's power), and revise the famous definition of strong-anonymity from the literature. Furthermore, we provide a technique to verify that a distributed protocol satisfies some of the proposed definitions. In Chapter 5 we provide (counterexample-based) techniques to debug complex systems, allowing for the detection of flaws in security protocols. Finally, in Chapter 6 we briefly discuss extensions to the frameworks and techniques proposed in Chapters 3 and 4.